Binomial Theorem Questions

We provide binomial theorem practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on binomial theorem skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

List of binomial theorem Questions

Question No Questions Class
1 The number of integral terms in the expansion of ( left(5^{1 / 2}+7^{1 / 8}right)^{1024} ) is 11
2 Coefficient of ( boldsymbol{x}^{k},(mathbf{0} leq boldsymbol{k} leq boldsymbol{n}) ) in
expansion of ( boldsymbol{P}=mathbf{1}+(mathbf{1}+boldsymbol{x})+(mathbf{1}+ )
( boldsymbol{x})^{2} ldots ldots+(1+boldsymbol{x})^{n} )
A. ( ^{n} C_{k} )
B. ( ^{n+1} C_{n-k-1} )
c. ( ^{n} C_{n-k} )
D. ( ^{n+1} C_{k+1} )
11
3 Find the coefficient of ( x^{-7} ) in the
expansion of ( left(boldsymbol{a} boldsymbol{x}-frac{mathbf{1}}{boldsymbol{b} boldsymbol{x}^{2}}right)^{11} )
( begin{array}{ll}text { A. } & ^{11} C_{5}end{array} )
В. ( ^{10} C_{4} )
c. ( ^{11} C_{4} )
D. ( ^{10} C_{5} )
11
4 Number of irrational terms in the
expansion of ( left(5^{frac{1}{6}}+2^{frac{1}{8}}right)^{100} )
A . 96
B. 97
c. 98
D. 99
11
5 The sum of the coefficient of first 3
terms in the expansion ( left(x-frac{3}{x^{2}}right)^{m} ) in
( 559 . ) Find the term of the expansion containing ( boldsymbol{x}^{mathbf{3}} )
11
6 If ( A ) and ( B ) are coefficients of ( x^{n} ) in the
expansions of ( (1+x)^{2 n} ) and ( (1+x)^{2 n-1} )
respectively, then ( frac{A}{B} ) is equal to
A .4
B . 2
( c .9 )
D. 6
11
7 The coefficient of three consecutive
terms in the expansion of ( (1+a)^{n} ) are
in ratio 1: 7: 21 , then find the value of
( boldsymbol{n} )
11
8 If ( C_{r} ) denotes the binomial coefficient
( ^{n} C_{r} ) then ( (-1) C_{0}^{2}+2 C_{1}^{2}+5 C_{2}^{2}+ )
( ldots(3 n-1) C_{n}^{2}= )
A ( cdot(3 n-2)^{2 n} C_{n} )
( ^{text {В }} cdotleft(frac{3 n-2}{2}right)^{2 n} C_{n} )
c. ( (5+3 n)^{2 n} C_{n} )
D ( cdotleft(frac{3 n-5}{2}right)^{2 n} C_{n+1} )
11
9 Find ( a ) if the ( 17^{t h} ) and ( 18^{t h} ) term of the
expanse on ( (2+a)^{50} ) are equal.
11
10 V1V9 VO101010101 011
The coefficient ofx7 in the expansion of (1-x-x2 + x) is
[2011]
(a) -132
(b) -144
() 132
(d) 144
11
11 The sum of all the coefficient of those
terms in the expansion of ( (a+b+c+d)^{8} ) which contains ( b ) but
not ( boldsymbol{c} ) is
( mathbf{A} cdot 6305 )
B ( cdot 4^{8}-3^{8} )
C. Number of ways of forming 8 digit numbers using digits 1,2,3 each number as atleast one 3
D. Number of ways of forming 4 digit numbers using digits 1,2,3 each number as atleast one 3
11
12 ( frac{C_{0}}{1}+frac{C_{1}}{2}+frac{C_{2}}{3}+ldots ldots+frac{C_{10}}{11}= )
A ( cdot frac{2^{11}}{11} )
B. ( frac{2^{11}-1}{11} )
c. ( frac{3^{11}}{11} )
D. ( frac{3^{11}-1}{11} )
11
13 (a)
4
(b)
120
25.
If the coefficents of x3 and x4 in the expansion of
powers of x are both zero, then
(a, b) is equal to:
[JEE M 2014]
(a) (14,272)) (10,272) (c) (16,251) (a) (14,251)
11
14 Number of rational term is the
expansion of ( left(7^{1 / 3}+11^{1 / 9}right)^{729} )
( A cdot 81 )
B. 82
c. 730
D. None of these
11
15 If ( (1+x)^{n}=sum_{i=0}^{n} C_{i} x^{i}, ) then the sum of
the products of ( C_{i} ) ‘s taken two at a time is represented by ( sum_{0 leq i leq j leq n} C_{i} C_{j} )
A ( cdot 2^{n}-frac{(2 n) !}{2(n !)^{2}} )
B. ( 2^{n}+frac{(2 n) !}{2(n !)^{2}} )
c. ( frac{1}{2}left(2^{2 n}+frac{(2 n) !}{(n !)^{2}}right) )
D. ( frac{2^{2 n}}{2(n !)^{2}} )
11
16 The middle term in the expansion of ( left(frac{a}{x}+b xright)^{12} )
A ( cdot 924 a^{6} b^{6} )
B. ( 924 a^{6} b^{5} )
( mathbf{c} cdot 924 a^{5} b^{5} )
D. ( 924 a^{5} b^{6} )
11
17 The number of rational terms in the
expansion of ( left(mathbf{9}^{1 / 4}+mathbf{8}^{1 / 6}right)^{1000} ) is:
A . 500
в. 400
( c .501 )
D. none of the above
11
18 A positive integer which is just greater ( operatorname{than}(1+0.0001)^{10000} ) is
( A cdot 3 )
B. 4
( c .5 )
D. 6
11
19 et n be positive integer. If the coefficients of 2nd, 3rd, and
1th terms in the expansion of (1 + x)” are in A.P., then the
value of n is …………
(1994 – 2 Marks)
11
20 Find ( A_{2}^{n}, ) if the fifth term of the
expansion of ( left(sqrt[3]{x}+frac{1}{x}right)^{n} ) does not
depend on ( boldsymbol{x} )
11
21 The coefficient of the middle term in the
expansion of ( (1+x)^{2 n} ) is
This question has multiple correct options
A ( cdot 2^{n} C_{n} )
в. ( frac{1.3 .5 ldots ldots(2 n-1)}{n !} 2^{text {। }} )
c. ( 2.6 ldots(4 n-2) )
D ( cdot 2.4 ldots ldots . .2 n )
11
22 The Coefficient of ( x^{n} ) in the expansion of
( (1+x)(1-x)^{n} ) is
A. ( (n-1) )
B ( cdot(-1)^{n-1} n )
C ( cdot(-1)^{n-1}(n-1)^{2} )
D・ ( (-1)^{n}(1-n) )
11
23 Find ( (boldsymbol{a}+boldsymbol{b})^{4}-(boldsymbol{a}-boldsymbol{b})^{4} ). Hence,
evaluate ( (sqrt{mathbf{3}}+sqrt{mathbf{2}})^{4}-(sqrt{mathbf{3}}-sqrt{mathbf{2}})^{4} )
11
24 If ( left(boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}+mathbf{1}right)^{boldsymbol{6}}=boldsymbol{a}_{boldsymbol{0}}+left(boldsymbol{a}_{boldsymbol{1}} boldsymbol{x}+frac{boldsymbol{b}_{boldsymbol{1}}}{boldsymbol{x}}right)+ )
( left(a_{2} x^{2}+frac{b_{2}}{x^{2}}right)+ldots+left(a_{6} x^{6}+frac{b_{6}}{x^{6}}right) ) the
find the value of ( a_{0} )
11
25 The coefficient of ( x^{3} ) in ( left(sqrt{x^{5}}+frac{3}{sqrt{x^{3}}}right)^{6} )
is
( mathbf{A} cdot mathbf{0} )
в. 120
( c cdot 420 )
D. 540
11
26 Which term in the expansion of ( (1+x)^{p} cdotleft(1+frac{1}{x}right)^{q} ) is independent of ( x )
where ( p, q ) are positive integers? What is the value of that term?
11
27 Find the coefficient of ( x^{-17} ) in the
expansion of ( left(x^{4}-frac{1}{x^{3}}right)^{15} )
A. 1200
B. -1331
c. -1365
D. -2016
11
28 Expand the following expression in
ascending powers of ( x ) as far as ( x^{3} ) ( frac{1+2 x}{1-x-x^{2}} )
11
29 ( (1+x)^{21}+(1+x)^{22}+ldots+(1+x)^{30} )
coefficient of ( boldsymbol{x}^{5} )
11
30 The term independent of ( x ) in the expansion of ( left(sqrt{frac{x}{3}}+frac{3}{2 x^{2}}right)^{10} ) will be
( ^{A} cdot frac{3}{2} )
в.
( c cdot frac{5}{2} )
D. None of these
11
31 13. If the coefficient of x’
equals the
(bx)
in
coefficient of x-
the relation
(a) a-b=1
, then a and b satisfy
[2005]
(b) a+b=1
(d) ab=1
11
32 26. The sum of coefficients of integral power of x in the binomial
expansion (1-27x) is:
[JEE M 2015
(b) (250 +1)
11
33 If the constant term in the binomial
expansion of ( left(x^{2}-frac{1}{x}right)^{n}, n quad epsilon quad N ) is 15
then the value of ( n ) is equal to
11
34 In the binomial expansion of ( (a- )
( b)^{n}, n geq 5, ) the sum of
5 th and 6 th terms is zero then a/b equal
to
A ( cdot frac{5}{n-4} )
B. ( frac{6}{n-5} )
c. ( frac{n-5}{6} )
D. ( frac{n-4}{5} )
11
35 Expand
(i) ( (sqrt{3}+sqrt{2})^{4} )
11
36 Binomial expansion of ( left(boldsymbol{x}^{k}+frac{mathbf{1}}{mathbf{2}^{mathbf{2 k}}}right)^{mathbf{3 n}} )
where ( n ) is a positive integer, always contains a term which is independent of
( mathbf{A} cdot x^{2} )
B. ( x )
( mathbf{c} cdot x^{3} )
D. none of the above
11
37 Expand the following binomial ( left(1-3 a^{2}right)^{6} ) 11
38 The coefficient of ( x^{4} ) in the expansion of
( left(frac{x}{2}-frac{3}{x^{2}}right)^{10} ) is equal to:
11
39 Write general term of this:-
( 2 xleft(3+2 x^{2}right)^{20} )
11
40 Solve : ( left(3 x-frac{1}{2 y}right)left(3 x+frac{1}{2 y}right) ) 11
41 The number of terms whose values
depends on ( x ) in the expansion of ( left(x^{2}-2+frac{1}{x^{2}}right)^{n} ) is
( mathbf{A} cdot 2 n+1 )
B. ( 2 n )
( c )
D. none of these
11
42 10. The coefficient of x^ in expansion of (1 + x) (1 – x)” is
(a) (-1)”-In
(b) (-1)” (1-n) [2004
(c) (−1)n-1(n-1) (d) (n-1)
11
43 The middle term of ( left(boldsymbol{x}-frac{1}{boldsymbol{x}}right)^{2 n+1} ) is
( mathbf{A} cdot^{2 n+1} C_{n} cdot x )
B. ( 2 n+1 C_{n} )
C ( cdot(-1)^{n} cdot 2^{2+1} C_{n} )
D ( cdot(-1)^{n} cdot^{2 n+1} C_{n} cdot x )
11
44 Coefficient of ( x^{5} ) in ( left(1+x^{2}right)^{5}(1+x)^{4} ) is
A . 60
B. 80
( c cdot 90 )
D. 100
11
45 Show that the middle term in the
expansion of ( (1+x)^{2 n} ) is
( frac{1.3 .5 ldots .(2 n-1)}{mid underline{n}} 2^{n} x^{n} )
11
46 The term independent of ( x ) in ( (1+ )
( x)^{n}left(1+frac{1}{x}right)^{n} ) is
A ( cdot C_{0}^{2}+2 cdot C_{1}^{2}+3 cdot C_{2}^{2}+ldots ldots ldots+(n+1) cdot C_{n}^{2} )
B. ( left(C_{0}+C_{1}+C_{2}+ldots ldots . .+C_{n}right)^{2} )
c. ( C_{0}^{2}+C_{1}^{1}+C_{2}^{2}+ldots ldots . .+C_{n}^{2} )
D. None
11
47 Find ( x, ) if it is known that the second
term of the expansion of ( left(x+x^{log x}right)^{5} ) is
equal to 1000000
11
48 Find the ( 13^{t h} ) term in the expansion of ( left(9 x-frac{1}{3 sqrt{x}}right)^{18} ) 11
49 Find the value of a given ( mathbf{3}+frac{mathbf{1}}{mathbf{4}}(mathbf{3}+boldsymbol{p})+frac{mathbf{1}}{mathbf{4}^{2}}(mathbf{3}+mathbf{2} boldsymbol{p})+frac{mathbf{1}}{mathbf{4}^{3}}(mathbf{3}+ )
( mathbf{3} boldsymbol{p})+ldots=? )
11
50 The term independent of ( x ) in the expansion of
( left(x^{2}+frac{1}{x}right)^{9} ) is
( A )
B. –
c. 48
D. 84
11
51 Expand ( left(x^{2}+frac{3}{x}right)^{4} ) 11
52 The ratio of fifth term from the
beginning to the fifth term from the end in the expansion of ( left(sqrt[4]{2}+frac{1}{sqrt[4]{3}}right)^{n} ) is
( sqrt{6}: 1 . ) If ( n=frac{20}{lambda}, ) find the value of ( lambda )
11
53 Find the coefficient of ( x^{5} ) in the product
( (1+2 x)^{6}(1-x)^{7} ) using binomial
theorem.
11
54 30. If the fourth term in the Binomial expansion of ( = + xlog8x
(x>0) is 20×87, then a value of x is:
(JEEM 2019-9 April (M)
(a) 8 (6) 8 (c) 8 (d) 82
11
55 The coefficient of ( x^{160} ) in the expansion
of ( left(x^{8}+right. )
1) ( ^{60}left(x^{12}+3 x^{4}+frac{3}{x^{4}}+frac{1}{x^{12}}right)^{-10} )
A. ( ^{30} C_{6} )
B. ( ^{30} C_{5} )
c. divisible by 189
D. divisible by 203
11
56 In the expansion of ( (x+sqrt{x^{2}-1})^{6}+ )
( (x-sqrt{x^{2}-1})^{6}, ) the number of terms is
A. 7
B. 14
( c cdot 6 )
D.
11
57 Which term in the expansion of ( left(frac{x}{3}-frac{2}{x^{2}}right)^{10} ) contains ( x^{4} ? )
( A )
B. 3
( c cdot 4 )
D.
11
58 Expand using Binomial Theorem ( left(1+frac{x}{2}-frac{2}{x}right)^{4}, x neq 0 ) 11
59 If rth term in the expansion of
( left(x^{2}+frac{1}{x}right)^{12} ) is independent of ( x, ) then
( boldsymbol{r}= )
( mathbf{A} cdot mathbf{9} )
B. 8
c. 10
D. none of these
11
60 Find the square of the following binomials by using the identity ( (-z+6) ) 11
61 The value of
( frac{18^{3}+7^{3}+3.18 .7 .25}{3^{6}+6.243 .2+15.81 .4+20.27 .8+15.9 .16+6.3 .32+64} )
( A cdot 4 )
( B .3 )
( c cdot 2 )
D.
11
62 The coefficient of ( x^{4} ) in the expansion of
( {sqrt{1+x^{2}}-x}^{-1} ) in ascending powers
of ( x, ) when ( |x|<1 ) is
( mathbf{A} cdot mathbf{0} )
в. ( frac{1}{2} )
( c cdot-frac{1}{2} )
D. ( -frac{1}{8} )
11
63 The sum of the last eight coefficients in
the expansion of ( (1+x)^{15}, ) is ( ? )
A ( cdot 2^{16} )
B . ( 2^{15} )
( c cdot 2^{14} )
D. none of these
11
64 if the coefficient of the middle term in
the expansion of ( (1+x)^{2 n+2} ) and ( p ) and
the coefficients of middle terms in the
expansion of ( (1+x)^{2 n+1} ) are ( q ) and ( r )
then
A ( . p+q=r )
в. ( p+r=q )
c. ( p=q+r )
D. ( p+q+r=0 )
11
65 If the first three terms in the expansion
of ( (1+a x)^{n} ) are ( 1,8 x, 24 x^{2} )
respectively, then ( a= )
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 4 )
D.
11
66 The coeff. of ( 8^{t h} ) term in the expansion of
( (1+x)^{10} ) is
A. 120
B. 7
c. ( ^{10} C_{8} )
D. 210
11
67 If the coefficient of 4 consecutive terms
in the expansion of ( (1+x)^{n} ) are
( a_{1}, a_{2}, a_{3}, a_{4} ) respectively, then show
that:
( frac{boldsymbol{a}_{mathbf{1}}}{boldsymbol{a}_{mathbf{1}}+boldsymbol{a}_{mathbf{2}}}+frac{boldsymbol{a}_{mathbf{3}}}{boldsymbol{a}_{mathbf{3}}+boldsymbol{a}_{boldsymbol{4}}}=frac{mathbf{2} boldsymbol{a}_{boldsymbol{3}}}{boldsymbol{a}_{mathbf{2}}+boldsymbol{a}_{boldsymbol{3}}} )
11
68 6.
mial expansion of (a – b)”, n25, the sum of the 5th
and 6th terms is zero. Then alb equals
(20015)
In the hi
(a)(n-5) 16
(0)55-4)
(6) [m-415
(d) Gen 5)
11
69 Find the coefficient of ( x^{50} ) in the
expression:
( (1+x)^{1000}+2 x(1+x)^{999}+ )
( mathbf{3} x^{2}(mathbf{1}+boldsymbol{x})^{mathbf{9} 9 mathbf{8}}+ldots .+mathbf{1 0 0} mathbf{1} boldsymbol{x}^{mathbf{1 0 0 0}} )
A ( .^{1000} mathrm{C}_{50} )
В. ( ^{1001} mathrm{C}_{50} )
( mathbf{c} cdot^{1002} C_{50} )
D. ( ^{1003} Omega_{50} )
11
70 In the expansion of ( left(sqrt[3]{4}+frac{1}{sqrt[4]{6}}right)^{20} )
This question has multiple correct options
A. the number of rational terms ( =4 )
B. the number of irrational terms ( =19 )
C. the middle term is irrational
D. the number of irrational terms ( =17 )
11
71 Expand ( (1-2 x)^{5} ) 11
72 r and n are positive integers r> 1, n > 2 and coefficient of
(r+2)th term and 3rth term in the expansion of (1 + x)2n are
equal, then n equals
[2002]
(2) 3r (6) 3r+1 (c) 2r (d) 2r+1
11
73 If the fourth term in the expansion of ( left(sqrt{frac{1}{x^{log x+1}}}+x^{1 / 12}right)^{6} ) is equal to 200
and ( x>1, ) then ( x ) is equal to
A ( cdot 10^{sqrt{2}} )
2 ( sqrt{2} )
B. 10
( c cdot 10^{4} )
D. None of these
11
74 Coefficient of ( boldsymbol{x}^{mathbf{5 0}} )
( (x>0), ) in ( (1+x)^{1000}+2 x(1+ )
( boldsymbol{x})^{999}+mathbf{3} boldsymbol{x}^{2}(mathbf{1}+boldsymbol{x})^{998}+ldots ) is
A. ( 1000 C_{50} )
B. ( ^{1000} C_{50} )
c. ( ^{1002} 250 )
D. ( ^{1000} C_{49} )
11
75 The coefficient of the middle term in the
binomial expansion in power of ( x ) of
( (1+alpha x)^{4} ) and of ( (1-alpha x)^{6} ) is the same
if ( alpha ) equals-
A ( cdot-frac{5}{3} )
в. ( frac{10}{3} )
( c cdot frac{-3}{10} )
D.
11
76 In the expansion of the expression ( (x+a)^{15}, ) if the eleventh term in the
geometric mean of the eighth and
twelfth terms, which term in the
expression is the greatest?
A. ( T_{6} )
в. ( T_{7} )
c. ( T_{8} )
D. ( T_{9} )
11
77 The ( 3 r d, 4 t h ) and 5 th terms in the
expansion of ( (1+x)^{n} ) are 60,160 and
240 respectively, then ( x= )
( A cdot 2 )
B. 4
( c .5 )
D. 6
11
78 Assertion
( (sqrt{2}-1)^{n} ) can be expressed as ( sqrt{N} ) ( sqrt{N-1} ) for ( forall N>1 ) and ( n in N )
Reason
( (sqrt{2}-1)^{n} ) can be written in the form ( boldsymbol{alpha}+boldsymbol{beta} sqrt{boldsymbol{2}} forall, boldsymbol{alpha}, boldsymbol{beta} ) are integers & n is a
positive integer.
A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,
B. Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion,
c. Assertion is true but Reason is false
D. Assertion is false but Reason is true.
11
79 The number of terms which are free
from radical signs in the expansion of ( left(y^{frac{1}{5}}+x^{frac{1}{10}}right)^{55} ) are
A. 5
B. 6
( c cdot 7 )
D. none of these
11
80 Find the cube of the following binomial expressions:
( frac{3}{x}-frac{2}{x^{2}} )
11
81 If ( |x|<1 ) then the coefficient of ( x^{n} ) in
expansion of ( left(1+x+x^{2}+x^{3} dotsright)^{2} ) is
( A )
B . ( n-1 )
( mathbf{c} cdot n+2 )
( mathbf{D} cdot n+1 )
11
82 If the constant term of the binomial zpansion ( left(2 x-frac{1}{x}right)^{n} ) is -160 , then ( n ) is equal to
A .4
B. 6
( c cdot 8 )
D. 10
11
83 The term in dependent of ( x ) in ( left(1+x+2 x^{3}right)left(frac{3 x^{2}}{2}-frac{1}{3 x}right)^{9} )
A ( cdot frac{25}{54} )
в. ( frac{17}{54} )
( c cdot frac{1}{6} )
D. ( -frac{17}{51} )
11
84 Find the 7 th term from the end in the
expansion of ( left(2 x^{2}-frac{3}{2 x}right)^{8} )
11
85 If ( n ) is an integer between 0 and 21 , then
the minimum value of ( n !(21-n) ! ) is
attained for ( n= )
A .
B. 10
c. 12
D. 20
11
86 The sum of the co-efficients of all odd
degree terms in the expansion of ( (x+sqrt{x^{3}-1})^{5}+(x+sqrt{x^{3}-1})^{5} )
( (x>1) ) is:
A . 2
B. –
( c cdot 0 )
( D )
11
87 Find the ( 13^{t h} ) terms in the expansion of
( left(9 x-frac{1}{3 sqrt{x}}right)^{18}, x neq 0 )
A . 18564
B. 87328
c. 17374
D. 35546
11
88 Write the general term in the expansion of ( left(x^{2}-y^{2}right)^{6} ) 11
89 The term independent of ( x ) in ( left(frac{1}{2} x^{frac{1}{3}}+right. )
( left.boldsymbol{x}^{frac{-1}{5}}right)^{8} ) is
A. ( frac{35}{8} )
B. 7
c. ( frac{7}{2} )
D . 28
11
90 Find the middle term(s) in the
expansion of :
( left(2 a x-frac{b}{x^{2}}right)^{12} )
A ( cdot frac{59136 a^{6} b^{6}}{x^{6}} )
В. ( frac{59163 a^{5} b^{5}}{x^{5}} )
c. ( frac{59631 a^{7} b^{7}}{x^{7}} )
D. None of these
11
91 Which number is larger (1.1) ( ^{100000} ) or
( mathbf{1 0}, mathbf{0 0 0} ? )
11
92 If ( a ) and ( b ) are distinct integers, prove
that ( a-b ) is a factor of ( a^{n}-b^{n} )
whenever ( n ) is a positive integer.
11
93 Find ( 7^{t h} ) term of ( left(frac{4 x}{5}-frac{5}{2 x}right)^{9} )
A. ( frac{10050}{x^{3}} )
в. ( frac{10500}{x^{3}} )
c. ( frac{1050}{x^{3}} )
D. ( frac{1000}{x^{3}} )
11
94 Given positive integers ( i>1, n>2 ) and that the coefficients of ( (3 r)^{t h} ) and ( (r+ )
2) ( ^{t h} ) terms in the bionomial expansion
of ( (1+x)^{2 n} ) are equal, then
A ( . n=2 r )
B. n=3r
c. ( n=2 r+1 )
D. None of these.
11
95 Given positive integers r>1, n >2 and that the coefficient of
(3r)th and (r + 2)th terms in the binomial expansion of
(1+ x)2n are equal . Then
(1983 – 1 Mark)
(a) n=2r
(c) n=2r+1
(c) n=3r
(d) none of these
11
96 The number of dissimilar terms in the
expansion of ( left(1-3 x+3 x^{2}-x^{3}right)^{20} ) is
A . 21
B. 32
c. 41
D. 61
11
97 The sum of the coefficients in the first
three terms of the expansion of ( left(x^{2}-frac{2}{x}right)^{m} ) is equal to ( 97 . ) Find the term of the expansion containing ( boldsymbol{x}^{4} )
11
98 In the expansion of ( (a+b)^{n}, ) the ratio of
the binomial coefficients of ( 2^{n d} ) and ( 3^{r d} )
terms is equal to the ratio of the binomial coefficients of ( 5^{t h} ) and ( 4^{t h} )
terms, then ( n= )
A . 4
B. 5
( c .6 )
D.
11
99 The term independent of ( x ) in ( (2 x- ) ( left.frac{1}{2 x^{2}}right)^{12} ) is
( mathbf{A} cdot-^{12} C_{3} cdot 2^{6} )
B. ( -^{12} C_{5} .2^{2} )
c. ( 12 C_{6} )
D. ( ^{12} C_{4} .2^{4} )
11
100 If ( n ) is a positive integer and ( (5 sqrt{5}+ )
( mathbf{1 1})^{2 n+1}=I+f ) where I is an integer
and ( 0<f<1 ) then
This question has multiple correct options
A. I is an even integer
B. ( (I+f)^{2} ) is divisible by ( 2^{2 n+1} )
c. lis divisible by 22
D. None of these
11
101 The coefficient of the term independent of ( x ) in the expansion of
( left(frac{x+1}{x^{frac{2}{3}}-x^{frac{1}{3}}+1}-frac{x-1}{x-x^{frac{1}{2}}}right)^{10} )
( A cdot 70 )
в. 112
c. 105
D. 210
11
102 Find the middle term in the expansion of ( left(frac{2 x^{2}}{3}-frac{3}{2 x}right)^{12} ) 11
103 If in the expansion of ( (1-x)^{2 n-1}, ) the
coefficient of ( x^{r} ) denoted by ( a_{r}, ) then :
( mathbf{A} cdot a_{r-1}+a_{2 n-r}=0 )
В ( cdot a_{r-1}-a_{2 n-r}=0 )
c. ( a_{r-1}+2 a_{a n-r}=0 )
D. None of these
11
104 If ( (1+x)^{10}=a_{0}+a_{1} x+a_{2} x^{2}+dots+ )
( a_{10} x^{10}, ) then the value of
( left(a_{0}-a_{2}+a_{4}-a_{6}+a_{8}-a_{10}right)^{2}+ )
( left(a_{1}-a_{3}+a_{5}-a_{7}+a_{9}right)^{2} ) is
( mathbf{A} cdot 2^{10} )
B . 2
( c cdot 2^{20} )
D. None of these
11
105 The ( 4 t h ) term from the end in the
expansion of ( left(frac{x^{3}}{2}-frac{2}{x^{2}}right)^{7} ) is
A . ( 35 x )
B. ( 70 x^{2} )
( c cdot 35 x^{2} )
D. ( 70 x )
11
106 The number of rational terms in the expansion of ( left(3 frac{1}{4}+7 frac{1}{6}right)^{144} )
A . 33
B. 23
( c cdot 12 )
D. 13
11
107 If the 20 th and 21 st terms in the
expansion of ( (1+x)^{40} ) are equal, then
the value of ( x ) is
A ( cdot frac{20}{21} )
в. ( frac{21}{20} )
c. 25
D. ( frac{1}{25} )
11
108 Find the term which has the exponent of ( x ) as 8 in the expansion of
( left(x^{frac{5}{2}}-frac{3}{x^{3} sqrt{x}}right)^{10} )
( mathbf{A} cdot T_{2} )
В. ( T_{3} )
c. ( T_{4} )
D. Does not exist
11
109 For ( boldsymbol{r}=mathbf{0}, mathbf{1}, mathbf{2}, mathbf{3}, dots, mathbf{1 0}, ) let ( boldsymbol{A}_{r}, boldsymbol{B}_{r}, boldsymbol{C}_{boldsymbol{r}} )
denote respectively the coefficient of ( x^{r} )
in the expansions of ( (1+x)^{10},(1+x)^{20} )
and ( (1+x)^{30} . ) Then ( sum_{r=1}^{10} A_{r}left(B_{10} B_{r}-right. )
( left.C_{10} A_{r}right) ) is equal to
( mathbf{A} cdot B_{10}-C_{10} )
B . ( A_{10}left(B_{10}^{2}right)-C_{10} A_{10} )
c.
D. ( C_{10}-B_{10} )
11
110 Number of irrational terms in the binomial expansion of ( left(3^{1 / 5}+7^{1 / 3}right)^{100} )
is
A . 94
B. 88
c. 93
D. 95
11
111 Show that the coefficient of ( a^{m} ) and ( a^{n} )
in the expansion of ( (1+a)^{m+n} ) are
equal.
11
112 Given that the ( 4^{t h} ) term in the expansion of ( left(2+frac{3 x}{8}right)^{10} ) has the maximum numerical value, then ( x ) can lie in the
interval(s)
This question has multiple correct options
( mathbf{A} cdotleft(2, frac{64}{21}right) )
B ( cdotleft(-frac{60}{23},-2right) )
( mathbf{C} cdotleft(-frac{64}{21},-2right) )
D ( cdotleft(2,-frac{60}{23}right) )
11
113 In any binomial expansion, the number
of terms are
( A cdot geq 5 )
B. ( geq 2 )
( c cdot geq 3 )
( D cdot geq 4 )
11
114 In the expansion of ( left(boldsymbol{a} sqrt{boldsymbol{a}}+frac{mathbf{1}}{boldsymbol{a}^{4}}right)^{boldsymbol{n}} ), the
coefficient in the second term exceeds
by 44 the coefficient in the first term. Find ( n )
A . 20
B . 25
( c .35 )
D. 45
11
115 If the sum of the coefficients of ( x^{2} ) and
coefficients of ( x ) in the expansion of
( (1+x)^{m}(1-x)^{n} ) is equal to ( -m, ) then
the value of ( 3(n-m) ) is
(Note ( : boldsymbol{m}, boldsymbol{n} text { are distinct }) )
11
116 Using Binomial theorem, evaluate ( (mathbf{9 9})^{5} ) 11
117 The product of two middle terms in the
expansion of ( left(frac{3 x^{2}}{2}-frac{1}{3 x}right)^{9} ) is
( ^{mathrm{A}} cdotleft(^{9} C_{4}right)^{2} cdot frac{x^{9}}{512} )
в. ( -9_{C_{4}} .^{9} C_{5}, frac{x^{8}}{512} )
c. ( _{-9}^{text {g }} q_{4} .^{9} C_{5} ). ( frac{x^{9}}{512} )
D・ ( _{9} C_{4} .^{9} C_{5}, frac{x^{9}}{256} )
11
118 Solve ( (1+i)^{4}+(1-i)^{4}= ) 11
119 Find the expansion of ( (boldsymbol{a}-boldsymbol{2} boldsymbol{x})^{boldsymbol{7}} ) 11
120 In the expansion of ( left(x^{3}-frac{1}{x^{2}}right)^{n}, n in N )
if the sum of the coefficient of ( x^{5} ) and
( x^{10} ) is ( 0, ) then ( n ) is
A . 25
B. 20
c. 15
D. None of these
11
121 Multiply the binomials.
( (y-8) ) and ( (3 y-4) )
11
122 Find the middle term in the expansion of ( (5 x-7 y)^{7} ) 11
123 15. If the expansion in powers of x of the function
is ao +ajx+azx? +azx?… then a, is
(1 – ax)(1-bx)
6″ -ah
b-a
[2006]
(b) a” – 6”
b-a
bn+1-an+1
b-a
an+l – 11+1
b-a
11
124 Let ( n ) be a positive integer such that
( left(1+x+x^{2}right)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+ )
( ldots+a_{2 n} x^{2 n}, ) then ( a_{r}= )
В ( cdot a_{2 n} ; 0 leq r leq 2 n )
D. None of these
11
125 The expansion ( left[boldsymbol{x}+left(boldsymbol{x}^{mathbf{3}}-mathbf{1}right)^{mathbf{1} / mathbf{2}}right]^{mathbf{5}}+[boldsymbol{x}- )
( left.left(x^{3}-1right)^{1 / 2}right]^{5} ) is a polynomial of degree
A. 5
B. 6
( c cdot 7 )
( D )
11
126 If the sum of the coefficients in the
expansion of ( (a+b)^{n} ) is ( 4096, ) then the
greatest coefficient in the expansion is
A ( cdot 924 )
в. 792
( c .1594 )
D. None of these
11
127 In the expansion of ( (1+x)^{n}, ) the ( 5^{t h} )
term is 4 times the ( 4^{t h} ) term and the ( 4^{t h} )
term is 6 times the ( 3^{r d} ) term. than ( n= )
( A cdot 9 )
B. 10
( c cdot 11 )
D. 12
11
128 Find the 7 th term from the end in the
expansion of ( left(9 x-frac{1}{3 sqrt{x}}right)^{18}, x neq 0 )
11
129 The number of terms with integral coefficients in the expansion of
( left(7^{1 / 3}+5^{1 / 2} cdot xright)^{600} ) is
A. 100
B. 50
( c .101 )
D. none of these
11
130 The coeffcient of ( x^{10} ) in the expansion of
( (1+x)^{2}left(1+x^{2}right)^{3}left(1+x^{3}right)^{4} ) is equal to
A . 52
B. 44
c. 50
D. 56
11
131 3. If (itaa)? 1 +880 +2438 t… thena…. andon… 11
132 Find the middle terms of the equation of
( left(x^{4}-frac{1}{x^{3}}right)^{11} )
( mathbf{A} cdot-462 x^{9}, 462 x^{2} )
B . ( -462 x^{8}, 462 x^{4} )
c. ( 462 x^{7},-462 x^{3} )
D. None of these
11
133 In the expansion of ( (sqrt{2}+sqrt[3]{5})^{20} ) the
number of rational terms will be:
( A cdot 3 )
B. 10
( c cdot 4 )
D.
11
134 Find the coefficient of ( x^{5} ) and ( x^{-15} ) in the
expansion of ( left(3 x^{2}-frac{1}{3 x^{3}}right)^{10} ? )
11
135 The total number of terms in the
expansion of ( (x+a)^{100}+(x-a)^{100} )
after simplification is
A .202
B. 51
( c .50 )
D. 49
11
136 Find the middle term(s) of ( left(frac{x^{3 / 2} y}{2}+right. )
( left.frac{2}{x y^{3 / 2}}right)^{13} )
11
137 Expand ( left(x^{2}+2 aright)^{5} ) by binomial
theorem.
11
138 Expand the binomial ( left(frac{2 x}{3}+frac{3 y}{2}right)^{20} u p ) to four terms. 11
139 7.
The number of integral terms in the expansion of
(13+ 5)256 is
[2003]
(a) 3 (6) 32 (6) 33 (d) 34
11
140 For ( mathbf{r}=mathbf{0}, mathbf{1}, dots, 10, ) let ( mathbf{A}_{mathbf{r}}, mathbf{B}_{mathbf{r}} ) and ( mathbf{C}_{mathbf{r}} )
denote, respectively, the coefficient of ( x^{r} ) in the expansions of ( (1+x)^{10},(1+ )
( mathbf{x})^{20} ) and ( (mathbf{1}+mathbf{x})^{30} ) Then ( sum_{r=1}^{10} boldsymbol{A}_{r}left(boldsymbol{B}_{10} boldsymbol{B}_{r}-right. )
( left.C_{10} A_{r}right) ) is equal to
A. ( mathrm{B}_{10}-mathrm{C}_{10} )
B . ( A_{10}left(B_{10}^{2}-C_{10} A_{10}right) )
c. 0
D. ( C_{10}-B_{10} )
11
141 Find the middle terms in the expansion
of ( (5 x-7 y)^{7} )
11
142 (0
)
10
10
Let S = { j (j – 1)!°C,, S, = $ 710c; and S3 = 2,2 10C
j=1
j=1
statement-1:S = 55 x 29.
Statement-2: S, =90 x 2
[2010]
nt-2: S = 90 x 28 and S. = 10 x 28.
Statement – 1 is true. Statement -2 is true ; Statement-2
not a correct explanation for Statement-1.
Statement -1 is true, Statement -2 is false.
Statement – 1 is false, Statement -2 is true.
statement – lis true, Statement 2 is true; Statement -2
1
1
.
c…datamant –
11
143 If ‘p’ and ‘q’ are the coefficients of ( x^{a} ) and
( x^{b} ) respectively in ( (1+x)^{a+b}, ) then
A. ( 2 p=q )
В. ( p+q=0 )
c. ( p=q )
D. ( p=2 q )
11
144 11.
For r=0, 1, …, 10, let A, B and C, denote, respectively,
the coefficient of x’ in the expansions of (1 + x)”, (2010)
10
(1 + x)20 and (1 + x)30. Then ZA(B10B.-C104,) is equal to
(a) B10-C10
(b) A10(B216C10410
(d) C10-B10
11
145 Find the middle term(s) in the
expansion of :
( left(3 x-frac{x^{3}}{6}right)^{9} )
A ( cdot frac{189}{8} x^{15},-frac{21}{16} x^{17} )
В ( cdot frac{189}{8} x^{17},-frac{21}{16} x^{19} )
C. ( frac{189}{7} x^{15},-frac{23}{13} x^{19} )
D. None of thes
11
146 The coefficient of ( boldsymbol{x}^{r}[mathbf{0} leq boldsymbol{r} leq boldsymbol{n}-mathbf{1}] ) in
the expression of ( (x+2)^{n-1}+(x+ )
2) ( ^{n-2} cdot(x+1)+(x+2)^{n-3} cdot(x+1)^{2}+ )
( ldots+(x+1)^{n-1} ) is
A ( cdot^{n} C_{r}left(2^{r}-1right) )
B. ( ^{n} C_{r}left(2^{n-r}-1right) )
c. ( ^{n} C_{r}left(2^{r}+1right) )
D. ( ^{n} C_{r}left(2^{n-r}+1right) )
11
147 la
J
in the expansion of (1 + x)” (1 – x)”, the coefficients of x
od r2 are 3 and – 6 respectively, then mis (1999 – 2 Marks)
a) 6 (6) 9
(c) 12 (d) 24
11
148 16.
b-a
For natural numbers m, nif (1-y)” (1 + y)”
=1+ay +a,y2 + ……. and a, = a, = 10, then (m, n) is
(a) (20,45)
(b) (35,20)
[2006]
(c) (45,35)
(d) (35,45)
ceth
11
149 Find the middle term of ( left(frac{a}{x}+frac{x}{a}right)^{10} ) 11
150 The coefficient of ( x^{30} ) in the expansion of
( left(1+2 x+3 x^{2}+dots .21 x^{20}right)^{2} ) is
A . 2706
в. 2450
( c .1481 )
D. 256
11
151 If it is known that the third term of the
binomial expansion ( left(x+x^{log _{10} x}right)^{3} ) is ( 10^{6} )
then ( x ) is equal to
A . 10
B. ( 10^{frac{5}{2}} )
c. 100
D. 5
11
152 The coefficient of ( x ) in ( left(x^{2}+frac{c}{x}right)^{5} ) is
A . 20
B. 10
( c cdot 10 c^{3} )
D. 20 ( c^{3} )
11
153 State the whether given statement is true or false
Prove that the coefficient of xnxn
A. True
B. False
11
154 28. The value of
(21C, -10C,)+(°C, – 10C,)+(!Cz – 1°C3)+(1C4 – 10C)
+…+(+1C70-10C10) is:
[JEE M 2017
(a) 220 -210
(b) 221 – 211
(c) 221 – 210
(d) 220 – 29
11111
11
155 If in the expansion of ( left(frac{1}{x}+x tan xright)^{5}, ) the ratio of ( 4^{t h} ) term to the ( 2^{n d} ) term is ( frac{2}{27} pi^{4} ) then the value of ( x ) can be
A ( cdot frac{-pi}{6} )
в. ( frac{-pi}{3} )
c.
D. ( frac{pi}{12} )
11
156 If the coefficients of ( a^{m} ) and ( a^{n} ) in the
expansion of ( (1+a)^{m+n} ) are ( alpha ) and ( beta ) then which one of the following is correct?
A ( cdot alpha=2 beta )
в. ( alpha=beta )
c. ( 2 alpha=beta )
11
157 The value of ( ^{n} C_{0}+3 times^{n} C_{1}+9 times^{n} )
( boldsymbol{C}_{2}+ldots+boldsymbol{3}^{n} times^{n} boldsymbol{C}_{n} )
A ( cdot 2^{n} )
B. ( 3^{n} )
( c cdot 4^{n} )
D. ( 5^{n} )
11
158 (1982 – 2 Marks)
The sum of the coefficients of the plynomial (1+x -3×2 2163
is ………
(1982-2 Marlo
IF(1 Iarn -119.242
11
159 The ( 3 r d, 4 t h, ) and 5 th terms in the
expansion ( (x+a)^{n} ) are respectively
( 84,280, ) and ( 560, ) find the values of ( x, a )
and ( n )
A. ( x=1, a=2, n=6 )
B. ( x=1, a=6, n=7 )
c. ( x=3, a=2, n=7 )
D. ( x=1, a=2, n=7 )
11
160 Find the term independent of ( ^{prime} x^{prime} ) in the expansion of the expression, ( (1+x+ )
( left.2 x^{3}right)left(frac{3}{2} x^{2}-frac{1}{3 x}right)^{9} )
11
161 ff ( f(x)=x^{4}+10 x^{3}+39 x^{2}+76 x+65 )
find the value of ( f(x-4) )
11
162 Find the coeffcient of:
( x^{-7} ) in the expansion of ( left(a x-frac{1}{b x^{2}}right)^{8} )
ii) ( x^{6} ) in the expansion ( left(a-b x^{2}right)^{10} )
11
163 If the fourth term in the expansion of
( (p x+1 / x)^{n} ) is ( 5 / 2 ) then the value of ( p )
is
( A )
B. 1/2
( c cdot 6 )
D.
11
164 The term independent of ( x ) in the expansion of ( left(1+x+2 x^{3}right)left(frac{3 x^{2}}{2}-right. )
( left.frac{1}{3 x}right)^{9} ) is
A ( cdot frac{13}{54} )
в. ( frac{15}{54} )
c. ( frac{17}{54} )
D. ( frac{19}{54} )
11
165 The coefficient of ( x^{5} ) in the expansion of
( left(1+x^{2}right)^{5}(1+x)^{4} ) is ( ? )
( mathbf{A} cdot 61 )
B. 59
c. zero
D. 60
11
166 The middle term of expansion of ( left(frac{10}{x}+frac{x}{10}right)^{10} )
A ( cdot^{7} C_{5} )
в. ( ^{8} C_{5} )
( mathrm{c} cdot^{9} mathrm{C}_{5} )
D. ( ^{10} C_{5} )
11
167 Find the term of expansion of ( left(x+frac{1}{x}right)^{n} ) which does not contain ( x ) 11
168 Let ( n ) be a positive integer. If the
coefficients of ( 2^{n d}, 3^{r d} ) and ( 4^{t h} ) terms in
the expansion of ( (1+x)^{n} ) are in A.P.
then the value of ( n ) is:
( mathbf{A} cdot mathbf{8} )
B. 27
c. 12
D.
11
169 Write the middle terms in the
expansion of ( left(frac{3 x}{7}-2 yright)^{10} )
11
170 ( (sqrt{3}+sqrt{2})^{4}-(sqrt{3}-sqrt{2})^{4}= )
A ( .20 sqrt{6} )
в. ( 30 sqrt{6} )
c. ( 5 sqrt{10} )
D. ( 40 sqrt{6} )
E ( .10 sqrt{6} )
11
171 The greatest value of the term independent for ( x ) in the expansion of
( left(x sin p+x^{-1} cos pright)^{10}, p in R, ) is
( A cdot 2^{5} )
в. ( frac{10 !}{(5 !)^{2}} )
c. ( frac{1}{2^{5}} cdot frac{10 !}{(5 !)^{2}} )
D. None of the above
11
172 distinct primes, then show that In nk ln2 (1984-2 Marks)
Find the sum of the series :
1 31 7
C – +- +- +-
Ar ….. up to m terms]
27 22r
Č (–19 “C,13+ 2 + 2 + y pu.. up to m terms]
237 x 15
24r ….. Un
11
173 |
(4)
O2
(U)
– 1
23. If n is a positive integer , then (13+1)?” -(13 – 1)?” is:
[2012]
(a) an irrational number
(6) an odd positive integer
an even positive integer
(d) a rational number other than positive integers
(C)
an even
11
174 Number of terms free from radical sign in the expansion of ( left(1+3^{1 / 3}+7^{1 / 2}right)^{10} )
is
( A cdot 4 )
B. 5
( c cdot 6 )
D.
11
175 Find the value(s) of k such that the term
independent of ( x ) in ( left(3 x^{2}+frac{k}{2 x}right)^{6} ) is 135
( A cdot pm 2 )
B. ±1
( c .pm 3 )
D. ±4
11
176 Let ( [x] ) denote the greatest integer part of a real number x. If
( boldsymbol{M}=sum_{n=1}^{40}left[frac{boldsymbol{n}^{2}}{mathbf{2}}right] )
then m equals
A . 5700
B. 5720
( c .5740 )
D. 11060
11
177 Sum of coefficients in the expeansion of
( (a+b+c)^{8} ) is
A. 2154
в. 6561
c. 729
D. 1944
11
178 The sum of the coefficients of first three
terms in the expansion of ( left(x-frac{3}{x^{2}}right)^{m}, x neq 0, m ) being a natura
number, is ( 559 . ) Find the term of the
expansion containing ( boldsymbol{x}^{mathbf{3}} )
11
179 Find the negative of middle term in the expansion of
( left(frac{2 x}{3}-frac{3}{2 x}right)^{6} )
11
180 The first 3 terms in the expansion of ( (1+a x)^{n}(n neq 0) ) are ( 1,6 x ) and ( 16 x^{2} )
Then the value of ( a ) and ( n ) are
respectively
A . 2 and 9
B. 3 and 2
c. ( 2 / 3 ) and 9
D. ( 3 / 2 ) and 6
11
181 If ( C_{0} . C_{1}, C_{2}, dots, C_{n} ) are the coefficients
of the expansion of ( (1+x)^{n}, ) then the value of ( sum_{0}^{n} frac{C_{k}}{k+1} ) is
A .
в. ( frac{2^{n}-1}{n} )
c. ( frac{2^{n+1}-1}{n+1} )
D. None of these
11
182 ( operatorname{Let}left(1+x^{2}right)^{2}(1+x)^{n}=A_{0}+A_{1} x+ )
( A_{2} x^{2}+ldots . ) If ( A_{0}, A_{1}, A_{2} ) are in A.P, then
the value of ( n )
( A cdot 2 )
B. 3
( c cdot 5 )
D. 7
11
183 The middle term in the expansion of
( left(frac{x}{y}+frac{y}{x}right)^{8} ) is
( A cdot^{8} C_{5} )
в. ( ^{8} mathrm{C}_{6} )
( mathrm{c} cdot^{8} mathrm{C}_{4} )
D. ( ^{8} mathrm{C}_{2} )
11
184 2
4
27. If the number of terms in the expansion of 1-+-
X+0, is 28, then the sum of the coefficients of all the term
in this expansion, is :
JJEEM 2016
(a) 243 (b) 729 (c) 64 (d) 2187
11
185 Find the middle term in the expansion
of :
( left(frac{x}{a}-frac{a}{x}right)^{10} )
11
186 Find the greater number in ( 300 ! ) and ( sqrt{300^{300}} ) 11
187 Find the number of terms in the
expansion of ( left(1-2 x+x^{2}right)^{7} )
11
188 The term independent of ( x ) in the expansion of ( left(x^{2}-frac{3 sqrt{3}}{x^{3}}right)^{10} ) 11
189 Using binomial theorem find the value of ( (102)^{3} ) 11
190 Find the term independent of ( x ) in ( (x+ ) ( left.frac{1}{x}right)^{4} ) 11
191 14. If x is so small that rand higher powers of x may be
(1 + x)2 –
1+
neglected, then
may be approximated as
(1-x)2
(a) 1-3×2
(b) 3x + 2×2
[2005]
8
12
to
8
11
192 6.
If x is positive, the first negative term in the expansion of
(1+x) 27/5 is
[2003]
(a) 6th term (b) 7th term (c) 5th term (d) 8th term.
11
193 The value of the expansion ( (sqrt{3}+1)^{5}+ )
( (sqrt{3}-1)^{5} )
( mathbf{A} cdot 88 )
B . 40
c. ( 88 sqrt{3} )
D. ( 40 sqrt{3} )
11
194 Find the second term of the binomial
expansion of ( left(sqrt[13]{boldsymbol{a}}+frac{boldsymbol{a}}{sqrt{boldsymbol{a}^{-1}}}right)^{m}, ) if ( boldsymbol{C}_{3}^{boldsymbol{m}} )
( C_{2}^{m}=4: 1 )
11
195 The third term from the end in the
expansion of ( left(frac{4 x}{3 y}-frac{3 y}{2 x}right)^{9} ) is
A ( cdot ) s ( _{C_{7}} frac{3^{5}}{2} frac{y^{5}}{x^{5}} )
В. ( -_{-9} sigma_{7} frac{3^{5}}{2^{3}} frac{y^{5}}{x^{5}} )
c. ( _{9} C_{7} frac{3^{5}}{2^{3}} frac{y^{5}}{x^{3}} )
D. none of these
11
196 Using Binomial Theorem, evaluate ( (101)^{4} ) 11
197 ( left(begin{array}{l}n \ 0end{array}right)+2left(begin{array}{l}n \ 1end{array}right)+2^{2}left(begin{array}{l}n \ 2end{array}right)+ldots .+2^{n}left(begin{array}{l}n \ nend{array}right) )
is equal to
A ( cdot 2^{n} )
B.
( c cdot 3^{n} )
D. None of these
11
198 If the coefficients of rth, (r+1)th, and (r+2)th terms in the
the binomial expansion of (1+y)” are in A.P., then mand
r satisfy the equation
[2005]
(a) m? – m(4r-1)+4r2 – 2 = 0
(b) m2 – m (4r+1)+4 r2 +2=0
(c) m2 – m (4r+1)+ 4 r2 – 2=0
(d) m? –m (4r-1)+4 2 +2 = 0
11
199 If in the expansion of ( (1+x)^{43}, ) the
coefficient of ( (2 r+1) t h ) term is equal
to coefficient of ( (r+2)^{t h} ) term. Find ( r ) ??
11
200 ( operatorname{Let}left(frac{2 x^{2}+x+2}{x}right)^{n}=sum_{r=m}^{r=t} a_{r} x^{r} )
then answer the following:
( fleft(a_{p}=a_{q} text { then } p+q=dotsright. )
11
201 The coefficient of ( x^{5} ) in the expansion of
( left(x^{2}-x-2right)^{5} ) is
A. 351
в. -82
c. -86
D. -81
11
202 If ( boldsymbol{X}=left{4^{n}-3 n-1: n in Nright} ) and
( boldsymbol{Y}={mathbf{9}(boldsymbol{n}-mathbf{1}): boldsymbol{n} in boldsymbol{N}}, ) where ( mathrm{N} ) is
the set of natural numbers, then ( boldsymbol{X} cup boldsymbol{Y} )
is equal to:
A. ( Y )
B. ( N )
c. ( Y-x )
D. ( x )
11
203 Write down and simplify:
The 25 th term of ( (5 x+8 y)^{30} )
11
204 The coefficient of ( x ) in the expansion of ( (1-a x)^{-1}(1-b x)^{-1}(1-c x)^{-1} ) is?
A ( . a+b+c )
B. ( a-b-c )
c. ( -a+b+c )
D. ( a-b+c )
11
205 Find the middle term in the expansion of ( left(frac{x}{a}-frac{a}{x}right)^{21} )
A ( cdot 20_{110} frac{x}{a},^{21} C_{10} frac{a}{x} )
В. ( _{20} C_{9} frac{x}{a},^{2} 16_{10} frac{a}{x} )
( ^{mathrm{C}} cdot_{21} C_{10} frac{x}{a^{2}},-^{21} 1_{10} frac{a}{x} )
D. ( _{21} C_{9} frac{x}{a},^{21} C_{10} frac{a}{x} )
11
206 f ( x+y=1, ) then ( sum_{r=0}^{n} r^{n} C_{r} x^{r} . y^{n-r}= )
( mathbf{A} cdot mathbf{1} )
B.
c. ( n x )
D. ( n y )
11
207 The total number of terms in the
expansion of ( (x+y)^{50}+(x-y)^{50} ) is
A . 51
B . 26
( c .102 )
D. 25
11
208 Show that the middle term in the
expansion of ( (1+x)^{2 n} ) is ( frac{1.3 .5 ldots . . .(2 n-1)}{n !} )
( 2^{n} x^{n} ; ) where ( n ) is a positive integer.
11
209 The third term from the end in the
expansion of ( left(frac{3 x}{5}-frac{5}{2 x}right)^{8} ) is
A ( cdot frac{35451}{15 x^{4}} )
в. ( frac{45455}{16 x^{4}} )
c. ( frac{39372}{15 x^{4}} )
D. ( frac{39375}{16 x^{4}} )
11
210 Expand the following binomial ( left(1-frac{1}{x}right)^{10} ) 11
211 ( operatorname{If}left(1+2 x+x^{2}right)^{n}=sum_{r=0}^{2 n} a_{r} x^{r}, ) then ( a_{r}= )
A ( cdotleft(^{n} C_{r}right)^{2} )
В. ( ^{n} C_{r} cdot^{n} C_{r+1} )
( c cdot^{2 n} C_{r} )
D. ( ^{2 n} C_{r+1} )
11
212 Number of irrational terms in the expansion of ( (sqrt{2}+sqrt{3})^{15} ) is equal to
A . 16
B. 7
c. 12
D. 15
11
213 The expansion of ( left[boldsymbol{x}+left(boldsymbol{x}^{mathbf{3}}-mathbf{1}right)^{mathbf{1} / 2}right]^{mathbf{5}}+ )
( left[x-left(x^{3}-1right)^{1 / 2}right]^{5} ) is a polynomial of
degree
( mathbf{A} cdot mathbf{8} )
B. 7
( c cdot 6 )
D. 5
11
214 In the expansion of ( left(3^{-x / 4}+3^{5 x / 4}right)^{n} )
the sum of binomial coefficient is 64
and term with the greatest bionomial coefficient term exceeds the third term
by ( (n-1) ) the value of ( x ) must be
( A cdot 0 )
B.
( c cdot 2 )
D. 3
11
215 Assertion
No three consecutive binomial
coefficient can be in G.P. & H.P.
Reason
Three consecutive binomial coefficients
are in A.P.
A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,
B. Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion,
c. Assertion is true but Reason is false
D. Assertion is false but Reason is true
11
216 Suppose that ( n ) is a natural number and
( boldsymbol{I}, boldsymbol{F} ) are respectively the integral part
and fractional part of ( (7+4 sqrt{3})^{n} . ) Then
show that
i) ( I ) is an odd integer
ii) ( (boldsymbol{I}+boldsymbol{F})(mathbf{1}-boldsymbol{F})=mathbf{1} )
11
217 If the 7 th and 8 th term of the binominal
expansion ( (2 a-3 b)^{n} ) are equal, then ( frac{2 a+3 b}{2 a-3 b} ) is to
A ( cdot frac{n-13}{n+1} )
в. ( frac{n+1}{13-n} )
c. ( frac{6-n}{13-n} )
D. ( frac{n-1}{13-n} )
11
218 In the binomial expansion of ( (1+y)^{n} )
where ( n ) is a natural number, the
coefficients of the ( 5^{t h}, 6^{t h} ) and ( 7^{t h} ) terms
are in A.P, find ( n )
This question has multiple correct options
( mathbf{A} cdot n=7 )
B . ( n=14 )
c. ( n=8 )
D. ( n=16 )
11
219 The value of
( left(10^{11}-10^{9}-2 times 11 times 10^{8}-3 times 11^{2} timesright. )
is equal to
( mathbf{A} cdot mathbf{0} )
B. ( 11^{10} )
( c cdot 11^{11} )
D. ( 10^{11} )
11
220 In the expansion of ( left(x-frac{1}{x}right)^{6} ), the
constant term is
A . -20
B . 20
c. 30
D. -30
11
221 The coefficient of the term independent of ( x ) in the expansion of ( left(frac{x+1}{x^{frac{2}{3}}-x^{frac{1}{3}}+1}-frac{x-1}{x-x^{frac{1}{2}}}right)^{10} )
A . 210
в. 105
c. 70
D. 112
11
222 If the fourth term in the expansion of
( left(sqrt{frac{1}{x log x+1}}+x frac{1}{12}right)^{6} ) is equal to
200 and ( x>1, ) then ( x ) is
A . 10
B. ( 10^{-4} )
( c cdot 1 )
D. -4
11
223 The co-efficient of ( x^{53} ) in the expression ( sum_{m=0}^{100} 100 c_{m}(x-3)^{100-m} 2^{m} ) is
B. ( 98_{c_{53}} )
( mathbf{c} .^{65} sigma_{53} )
D. ( 100 c_{65} )
11
224 The coefficient of ( t^{24} ) in the expansion of
( left(1+t^{2}right)^{12}left(1+t^{12}right)left(1+t^{24}right) ) is
A. ( ^{12} C_{6}+2 )
B. ( ^{12} C_{5} )
( mathbf{c} cdot^{12} C_{6} )
D. ( ^{12} C_{7} )
11
225 Find the ( (p+2) ) th term from the end in ( left(x-frac{1}{x}right)^{2 n+1} ) 11
226 The value of ( x ) in the expression
( left(x+x^{log _{10} x}right)^{5}, ) if the third term in the
expansion is ( 10,00,000, ) is This question has multiple correct options
A ( cdot 10^{-1} )
B . ( 10^{text {। }} )
( mathbf{c} cdot 10^{-5 / 2} )
D. ( 10^{5 / 2} )
11
227 For positive integers ( n_{1}, n_{2} ) the value of
the expression ( (1+i)^{n_{1}}+left(1+i^{3}right)^{n_{1}}+ )
( left(1+i^{5}right)^{n_{2}}+left(1+i^{7}right)^{n_{2}}, i=sqrt{-1} ) is a
real number if and only if
( mathbf{A} cdot n_{1}=n_{2}+1 )
В . ( n_{1}=n_{2}-1 )
c. ( n_{1}=n_{2} )
D ( . n_{1}>0, n_{2}>0 )
11
228 If ( boldsymbol{X}=left{mathbf{8}^{n}-mathbf{7} boldsymbol{n}-mathbf{1}, boldsymbol{n} in boldsymbol{N}right} ) and ( boldsymbol{Y}= )
( mathbf{4} 9(n-1), n in N, ) then ( (operatorname{given} n>1) )
A. ( X subset Y )
в. ( Y subset X )
c. ( X=Y )
D. ( X nsubseteq Y )
11
229 The positive integerjust greater than
( (1+0.0001)^{10000} ) is
( A cdot 4 )
B. 5
( c cdot 2 )
D.
11
230 The sum of the coefficients in the first,
second, and third terms of the expansion of ( left(x^{2}+frac{1}{x}right)^{m} ) is equal to 46 Find the term of the expansion which does not contain ( x )
11
231 If ( (1+a x)^{n}=1+8 x+24 x^{2}+dots ) then
( boldsymbol{a} times boldsymbol{n} ) is:
( mathbf{A} cdot mathbf{8} )
B. 12
c. 16
D . 24
11
232 Evaluate ( (sqrt{3}+sqrt{2})^{6}-(sqrt{3}-sqrt{2})^{6} ) 11
233 19. Let n be a positive integer and
(1994 – 5
(1 + x + x2)n = a+a, x+ …………+ a), x2
Show that a 2-a,2+ a,2 ………….+ a,,2=an
11
234 Find the expansion ( left(3 x^{2}-2 a x+3 a^{2}right)^{3} )
using binomial theorem.
11
235 The value of the term independent of
( x ) in the expansion of ( left(x^{2}-frac{1}{x}right)^{27} ) is :
( mathbf{A} cdot mathbf{9} )
B. 18
c. 48
D. 84
11
236 The number of integral terms in ( (sqrt{3}+sqrt[8]{2})^{64} ) is
( mathbf{A} cdot mathbf{8} )
B. 7
( c .9 )
D. 6
11
237 If for ( 1 leq m leq n, f(m, n)=C_{0}-C_{1}+ )
( C_{2}-ldots .(-1)^{m-1} C_{m-1}, ) find ( f(9,5) )
11
238 ( 5^{t h} ) term from the end in the expansion
of ( left(frac{x^{2}}{2}-frac{2}{x^{2}}right)^{12} ) is
B . ( 7920 x^{4} )
c. ( 7920 x^{-4} )
D. ( -7920 x^{4} )
11
239 The term independent of ( x ) in the expansion of ( left(x-frac{1}{x}right)^{4}left(x+frac{1}{x}right)^{3} ) is
A . -3
B.
( c .3 )
D.
11
240 Find the fourteenth term of ( (3-a)^{15} ) 11
241 If the coefficient of three consecutive
terms in the expansion of ( (1+x)^{n} ) be
( 165,330, ) and ( 462 . ) Find ( n )
11
242 ( (1-sqrt{2})^{6}= )
A ( .98-70 sqrt{2} )
В. ( 99-70 sqrt{2} )
D. ( 98+70 sqrt{2} )
11
243 9.
The coefficient of the middle term in the binomial expansion
in powers of x of (1+ ax)4 and of (1 – ax) is the same if a
equals
[2004]
(a)
5 (6)
(b)
(c) To (d) –
11
244 The coefficient of ( x^{9} ) in the expansion of ( left(x^{3}+frac{1}{2^{t}}right)^{11}, ) where ( t=log _{sqrt{2}}left(x^{frac{3}{2}}right) )
A . -5
в. 330
( c .520 )
D. ( 5+log _{sqrt{2}}(3) )
11
245 Find the term independent of ( x ) in the expansion of ( left(sqrt{frac{boldsymbol{x}}{mathbf{3}}}+frac{mathbf{3}}{mathbf{2} boldsymbol{x}^{2}}right)^{10} )
( mathbf{A} cdot T_{3} )
в. ( T_{4} )
c. ( T_{5} )
D. No term will be independent of ( x )
11
246 In the expansion of ( left(x^{3}-frac{1}{x^{2}}right)^{2}, ) where n is a positive integer, the sum of the
coefficients of ( boldsymbol{x}^{boldsymbol{6}} ) is ( mathbf{1} )
11
247 The 2 nd, 3 rd and 4 th terms in the
expansion of ( (x+y)^{n} ) are 240,720,1080 respectively; find ( x, y, n )
11
248 Solve ( :left(frac{2 n}{2 n-1}right)^{p}=left(frac{1}{1-left(frac{p}{2 n}right)}right)^{p} ) 11
249 ( f^{n-1} C_{r}=left(k^{2}-3right)^{n} C_{r+1}, ) then ( k )
B. ( [2, infty) )
c. ( [-sqrt{3}, sqrt{3}] )
D. ( (sqrt{3}, 2] )
11
250 In the expansion of ( left(x^{2}-frac{1}{4}right)^{n}, ) the coefficient of third term is ( 31, ) then the
value of ( n ) is-
A . 30
B. 31
c. 29
D. 32
11
251 ff ( left(x^{2}+frac{1}{x}right)^{n} ) has exactly one middle term which is equal to ( alpha . x^{3} ) then the
value of ( (boldsymbol{alpha}+boldsymbol{n}) ) is- ( quad(boldsymbol{n} in boldsymbol{N}) )
A . 18
B . 21
c. 24
D. 26
11
252 3.
The positive integer just greater than (1 +0.0001)10000 is
[2002]
(2) 4 (6) 5 (c) 2 (d) 3
11
253 Expand
( left(2 x^{2}+3right)^{4} )
11
254 If ( (5+2 sqrt{6})^{n}=m+f ), where ( n ) and ( m )
are positive integers and ( 0 leq f<1 )
then
( frac{1}{1-f}-f ) is equal to
A ( cdot frac{1}{m} )
в. ( m )
c. ( _{m+frac{1}{m}} )
D. ( _{m-} frac{1}{m} )
11
255 If the third term in the expansion of ( left[x+x^{log _{10} x}right]^{5} ) is ( 10^{6}, ) then ( x ) can be
This question has multiple correct options
A ( cdot 10^{-1 / 3} )
B. 10
c. ( 10^{-5 / 2} )
D. ( 10^{2} )
11
256 The middle term in the expansion of
( left(frac{2 x}{3}-frac{3}{2 x^{2}}right)^{2 n} ) is
A ( cdot 2^{n} mathrm{C}_{n} )
B cdot ( (-1)^{n}left[(2 n !) /(n !)^{2}right] cdot x^{-n} )
( mathrm{c} cdot_{2 n} mathrm{C}_{n} cdot frac{1}{x^{n}} )
D. none of these
11
257 Expansion of ( (boldsymbol{y}+boldsymbol{x})^{n} ) is 11
258 The number of rational terms in the
expansion of ( left(x^{frac{1}{5}}+y^{frac{1}{10}}right)^{45} ) is
A. 5
B. 6
( c cdot 4 )
D.
11
259 If ( T_{r} ) denotes the ( r^{t h} ) term in the
expansion of ( left(x+frac{1}{x}right)^{23}, ) then
( mathbf{A} cdot T_{12}=T_{13} )
В . ( x^{2} . T_{13}=T_{12} )
c. ( x^{2} . T_{12}=T_{13} )
D. ( T_{12}+T_{13}=25 )
11
260 e sum of the co-efficients of all odd degree terms in the
expansion of
(x+Vx3 -1)3 +(x-Vx3-1)”,(x>1) is : [JEEM 2018]
(a) o (6) 1 do (c) 2
(d) – 1
11
261 The middle term in the expansion of ( left(x+frac{1}{x}right)^{10} )
A ( cdot ) io ( _{1} frac{1}{x} )
в. ( ^{10} C_{5} )
c. ( ^{10} C_{6} )
D. ( ^{10} C_{7} x )
11
262 Find the middle term in the expansion of ( left(frac{2 x}{3}+frac{3}{2 x}right)^{10} )
A . 210
в. 630
( c .252 )
D. 756
11
263 UL. TUU lay uuu
23.
Let n be any positive integer. Prove that
(1999 – 10 Marks)
2n-k
mlk
n-k)
2n-2m
(2n-4k+1) 9n-2k =
on-Ak
(2n-2k +1) (2n – 2m)
In-m
kao
for each non-be gatuve integer msn. | Here
11
264 Find the sixth term in the expansion of
( left(2 x^{2}-frac{3}{7 x^{3}}right)^{11} )
A ( cdot-^{11} C_{5} frac{2^{6} 3^{5}}{7^{5}} x^{3} )
В ( cdot quad^{11} C_{5} frac{2^{6} 3^{5}}{7^{5}} x^{-3} )
c. ( _{-11} C_{5} frac{2^{6} 3^{5}}{7^{5}} x^{-3} )
D. None of these
11
265 Find coefficient of ( a^{3} b^{4} c^{5} ) in the
expansion of ( (b c+c a+a b)^{6} )
11
266 In the expansion of ( left(7^{1 / 3}+11^{1 / 9}right)^{6561} )
prove that three will be only 730 term which are free from radicals
11
267 In the expansion of ( (1+x)^{n}, ) the
binomial coefficients of 3 consecutive
terms are respectively 220,495 and 792
then ( n= )
A .4
B. 8
( c cdot 12 )
D. 16
11
268 If the constant term of the binomial expansion ( left(2 x-frac{1}{x}right)^{n} ) is ( -160, ) then ( n ) is equal to –
A .4
B. 6
c. 8
D. 10
11
269 Expand to 4 terms the following expressions:
( (1+x)^{frac{2}{5}} )
11
270 f ( log 1001=3.000434 ), find the number of digits in ( 1001^{101} ) 11
271 Expansion of ( (3 x+2)^{3} ) is ( 27 x^{3}+8+ )
( 18 x(3 x+2) )
A. True
B. False
11
272 The tenth term in the expansion of
( left(2 x^{2}+frac{1}{x}right)^{12} )
A ( cdot frac{1760}{x^{3}} )
в. ( -frac{1760}{x^{3}} )
c. ( frac{1760}{x^{2}} )
D. none of the above
11
273 3
17.
145, 35)
In the binomial expansion of (a – b)”, n > 5, the sum of
[2007]
and 6th terms is zero, then a/b equals
n-5 a n-4 o 5 (d)
(a)”
(6) “5* ©n-4
n-5
20071
11
274 Write general terms of this
( 2 xleft[3+2 x^{2}right]^{20} )
11
275 The coefficient of ( x^{11} ) in the expansion of
( left(1-2 x+3 x^{2}right)(1+x)^{11} ) is
A . 164
в. 144
c. 116
D. none of these
11
276 The coeficient of x’in (
The coefficient of x4 in
is
(1983 – 1 Mark)
“(19831 Mar
(b) 504
504
(a)
405
256
259
450
263
(d) none of these
11
277 1200T
18.
The sum of the series
20 Co – 20G + 2002 – 2003 +… -.+ 206, is
(a) o (6) 20. c) _2000 (a) 220610
11
278 Expand ( left(boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}right)^{boldsymbol{6}} ) 11
279 Find the middle term(s) in the
expansion of ( left(1+3 x+3 x^{2}+x^{3}right)^{2 n} )
11
280 If ( left(1+x^{2}right)^{2}(1+x)^{n}=C_{0}+C_{1} x+ )
( C_{2} x^{2}+cdots, ) and if ( C_{0}, C_{1}, C_{2} ) are in A.P.
find ( n )
This question has multiple correct options
A .2
B. 3
( c cdot 4 )
D. 5
11
281 The middle term in the expansion of
( left(x+frac{1}{x}right)^{10} )
A ( cdot ) io ( C_{4} cdot frac{1}{x} )
в. ( ^{10} C_{5} )
c. ( _{10} mathrm{C}_{5} . frac{1}{x} )
D. ( ^{10} C_{6} . x )
11
282 The greatest value of the term independent of ( x, ) as ( alpha ) varies over ( R ) in
the expansion of ( left(x cos alpha+frac{sin alpha}{x}right) ) is
B. ( ^{20} C_{19} )
c. ( 20 C_{6} )
D. ( ^{20} C_{6}left(frac{1}{2}right)^{10} )
11
283 If the sum of the coefficients in the expansion of (a + b)” is
4096, then the greatest coeficient in the expansion is
[2002]
(a) 1594 (b) 792 (c) 924 (d) 2924
anni 10000.
11
284 Find the middle term in the expansion
of ( left(1-2 x+x^{2}right)^{n} )
A ( cdot frac{(2 n) !}{(n !)^{2}}(-1)^{n} x^{n} )
B. ( frac{(2 n) !}{(n !)}(-1)^{n} x^{2} n )
c. ( frac{(2 n) !}{(n !)^{2}} x^{n} )
D. None of these
11
285 Find the coefficients of ( x^{2} ) and ( x^{3} ) in the
expansion of ( (2-x)^{6} )
11
286 Write the general term in the expansion
of ( left(x^{2}-yright)^{6} )
11
287 The ratio of coefficient of ( x^{3} ) and ( x^{4} ) in
expansion ( (1+x)^{12} ) is:
( A cdot frac{4}{9} )
B . ( frac{1}{3} )
( c cdot frac{2}{3} )
D.
11
288 και
η 5. Σε
αυτά της – Σ
ε
εθει με τον ερχολ το
ΥΞΟ
2n -1
(2004)
(c)
n – 1
11
289 Which of the following binomial expressions has a least term
independent of ( x ? )
( ^{mathrm{A}} cdotleft(sqrt{x}-frac{3}{x^{2}}right)^{1} )
B. ( left(x+frac{1}{x}right)^{6} )
c. ( (1+x)^{32} )
( ^{mathrm{D}}left(frac{3}{2} x^{2}-frac{1}{3 x}right)^{9} )
11
290 Find the coefficient of ( x^{17} ) in ( (x+y)^{20} ? ) 11
291 If the middle term in the expansion of ( left(x^{2}+frac{1}{x}right)^{n} ) is ( 924 x^{6}, ) then ( n= )
A . 10
B. 12
c. 14
D. none of these
11
292 The term independent of ( x ) in the expansion of ( left(boldsymbol{x}^{2}-frac{mathbf{1}}{boldsymbol{x}}right)^{boldsymbol{6}} ) is
A . -12
B. 15
( c cdot 24 )
D. -15
11
293 The coefficient of ( t^{50} ) in
( left(1+t^{2}right)^{25}left(1+t^{25}right)left(1+t^{40}right)left(1+t^{45}right)(1 )
is
A ( cdot 1+sqrt[25]{5} )
B. ( 1+^{25} C_{5}+^{25} C_{7} )
( mathbf{C} cdot 1+^{25} C_{7} )
D. None of these
11
294 What is the unit digit in the product ( left(3^{65} times 6^{59} times 7^{71}right) ? )
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 4 )
D. 6
11
295 The middle term in the expansion of ( left(1-3 x+3 x^{2}-x^{3}right)^{2 n} ) is
( mathbf{A} cdot 6 n_{C_{3 n}}(-x)^{3 n} )
B . ( 6 n_{C_{2 text { n }}}(-x)^{2 n+1} )
( mathbf{c} cdot 4 n_{C_{S n}}(-x)^{3 n} )
D. ( 6 n_{C_{3 n}}(-x)^{3 n-1} )
11
296 The middle term in the expansion of
( (1+x)^{2 n} ) is
A. ( frac{1.3 .5 ldots(2 n-1)}{n} x^{n} )
B. ( frac{1.3 .5 ldots(2 n-1)}{n !} 2^{n-1} x^{n} )
c. ( frac{1.3 .5 ldots(2 n-1)}{n !} x^{n} )
D. ( frac{1.3 .5 ldots(2 n-1)}{n !} 2^{n} x^{n} )
11
297 Find the middle term in the expansion
of
( left(frac{2}{3} x-frac{3}{2 x}right)^{20} )
A ( .^{20} C_{10} x^{10} y^{10} )
B. ( ^{20} C_{11} x^{11} y^{11} )
C. ( ^{20} C_{9} x^{11} y^{10} )
D. None of these
11
298 f ( ^{n} C_{4},^{n} C_{5},^{n} C_{6} ) of ( (1+x)^{n} ) are in A.P.
then ( n= )
A . 12
B. 1
( c cdot 7 )
( D .8 )
11
299 Expand the expression ( (2 x-3)^{6} ) 11
300 Coefficient of ( boldsymbol{x}^{boldsymbol{9}} ) in ( rightarrow(mathbf{1}+boldsymbol{x})(mathbf{( 1}+boldsymbol{t}) )
( left.left.boldsymbol{x}^{2}right)left(mathbf{1}+boldsymbol{x}^{3}right) ldots ldotsleft(boldsymbol{1}+boldsymbol{x}^{mathbf{1 0 0}}right)right) ? )
11
301 14. F E 4, (x-2)* = 6, (x – 3)” and ax = 1 for all
p=0
(1992 – 6 Marks)
k > n, then show that b, = 2NFC
11
302 19.
Statement-1:
“C, = (n + 2)2n-1
[2008]
r=0
Statement-2: E(r+1) “Cyx” = (1 + x)” + nx(1+x)”-.
r=0
(a) Statement-1 is false, Statement-2 is true
(b) Statement-1 is true, Statement-2 is true; Statement -2 15
a correct explanation for Statement-1
Statement -1 is true, Statement-2 is true; Statement -2
is not a correct explanation for Statement-1
(d) Statement -1 is true, Statement-2 is false
11
303 Find the term of the expansion of ( (sqrt[3]{x^{-2}}+x)^{7} ) containing ( x ) in the
second power
( mathbf{A} cdot T_{4} )
в. ( T_{5} )
c. ( T_{6} )
( mathbf{D} cdot T_{7} )
11
304 The number of non-zero terms in the
expansion of ( (sqrt{7}+1)^{75}-(sqrt{7}-1)^{75} )
is
A . 36
B. 37
c. 38
D. 39
11
305 ( 9^{t h} ) term in the expansion of
( left(frac{x}{a}-frac{3 a}{x^{2}}right)^{12} )
A ( cdot^{12} C_{9} cdot 3^{9} x^{-12} a^{6} )
В. ( ^{12} C_{6} cdot 3^{8} x^{-16} a^{6} )
C ( .^{12} C_{4} cdot 3^{8} x^{-12} a^{4} )
D. none of the above
11
306 24.
The term independent of x in expansion of
J
JEEM 2013
x+1_ -_*-1 is
(r2/3 – X1/3+1 x – x12
(a) 4
(b) 120
(C) 210
(d) 310
11
307 The middle term in the expansion of
( left(frac{10}{x}+frac{x}{10}right)^{10} )
A . ( ^{10} C_{5} )
в. ( ^{10} C_{6} )
c. ( _{10} mathrm{C}_{5} frac{1}{x^{10}} )
D. ( ^{10} C_{5} x^{10} )
E . ( ^{10} C_{5} 10^{10} )
11
308 Find the sum of the coefficients of the
terms of the expansion ( left(1+x+2 x^{2}right)^{6} )
11
309 Show that the middle term in the
expansion of ( (1+x)^{2 n} ) is
( frac{1.3 .5 ldots(2 n-1)}{n !} 2^{n} x^{n}, ) where ( n ) is a
positive integer.
11
310 The value of ( (sqrt{5}+1)^{5}-(sqrt{5}-1)^{5} ) is:
A . 252
в. 352
c. 452
D. 552
11
311 Prove that the coefficient of ( x^{n} ) in the
expression of ( (1+x)^{2 n} ) is twice the
coefficient of ( x^{n} ) in the expression of
( (1+x)^{2 n-1} )
11
312 The co-efficient of ( x^{5} ) in the expansion of
( (1+x)^{21}+(1+x)^{22}+dots dots+ )
( (1+x)^{30} ) is:
A . ( ^{51} C_{5} )
в. ( ^{9} C_{5} )
c. ( ^{31} C_{6}-^{21} C_{6} )
D. ( ^{30} C_{5}+^{20} C_{5} )
11
313 Find the value ( (sqrt{3}+1)^{4}+(sqrt{3}-1)^{4}=? ) 11
314 For ( boldsymbol{r}=mathbf{0}, mathbf{1}, mathbf{2},, dots mathbf{1 0} ) let ( boldsymbol{A}_{r}, boldsymbol{B}_{r} ) and ( boldsymbol{C}_{boldsymbol{r}} )
denote respectively the coefficient of ( boldsymbol{x}^{r} ) in the expansions of ( (1+x)^{10},(1+x)^{20} ) and ( (1+x)^{30} . ) Then
( sum_{r=1}^{10} A_{r}left(B_{10} B_{r}-C_{10} A_{r}right) ) is equal to
A. ( B_{10}-C_{10} )
B . ( A_{10}left(B_{10}^{2}-C_{10} A_{10}right) )
c.
D. ( C_{10}-B_{10} )
11
315 If the second term of the expansion ( left[boldsymbol{a}^{1 / 13}+frac{boldsymbol{a}}{sqrt{boldsymbol{a}^{-1}}}right]^{n} quad boldsymbol{i s} 14 boldsymbol{a}^{5 / 2}, ) then the
value of ( frac{n_{mathbf{S}}}{n_{mathbf{C}_{2}}} ) is
A .4
B. 3
c. 12
D. 6
11
316 In the expression of ( left(2^{x}+frac{1}{4^{x}}right)^{n} ) ratio of
2nd and third terms is given by ( t_{3} / t_{2}= )
7 and the sum of the co-efficients of 2 nd
and ( 3 mathrm{rd} ) term is ( 36, ) then the value of ( x )
is
A ( frac{-1}{3} )
в. ( frac{-1}{2} )
( c cdot frac{1}{3} )
D.
11
317 If ( omega neq 1 ) is a cube root of unity and
( (omega+x)^{n}=1+12 omega+69 omega+ldots . ) then
values of ( 4 n ) and ( omega ) respectively are
A . 36,1
в. 12,2
c. ( 24,1 / 2 )
D. ( 18,1 / 3 )
11
318 In the expansion of ( left(7^{1 / 3}+11^{1 / 9}right)^{6561} ) prove that there will be only 730 terms which are free from radicals. 11
319 If the coefficients of ( x ) and ( x^{2} ) in the
expansion of ( (1+x)^{m}(1-x)^{n} ) are 3
and -6 respectively. Find the values of
( m ) and ( n )
11
320 The coefficient of ( t^{4} ) in the expansion of
( left(1+t^{2}right)^{3} )
( A )
B. 3
( c cdot-3 )
D. –
11
321 The highest term in the expansion of ( (2 sqrt{5}+sqrt[6]{7})^{6} ) is
A. ( 800 sqrt{35} )
55
в. ( 700 sqrt{35} )
c. ( 320 sqrt{5} )
D. ( 100 sqrt{7} )
11
322 If ( boldsymbol{f}(boldsymbol{x})=(mathbf{1}+boldsymbol{x})^{15}=boldsymbol{C}_{0}+boldsymbol{C}_{1} boldsymbol{x}+ )
( C_{2} x^{2}+ldots+C_{15} x^{15}, ) then ( f(2)= )
( mathbf{A} cdot 1^{15} )
B. ( 3^{15} )
( c cdot 2^{15} )
D. None of these
11
323 Consider the expansion ( left(x^{2}+frac{1}{x}right)^{15} ) Consider the following statements:
1. There are 15 terms in the given expansion.
2. The coefficient of ( x^{12} ) is equal to that
of ( x^{3} )

Which of the statements is/are correct
( ? )
A. 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor 2

11
324 Find the ( 4^{t h} ) term in the expansion of
( (x-2 y)^{12} )
11
325 If ( c_{0}, c_{1}, c_{2}, ldots ldots c_{n} ) are the coefficients in
the expansion of ( (1+x)^{n}, ) when ( n ) is a
positive integer, prove that
(1) ( c_{0}-c_{1}+c_{2}-c_{3}+dots dots+ )
( (-1)^{r} c_{r}=(-1)^{r} frac{mid n-1}{|underline{r}| n-r-1} )
(2) ( c_{0}-2 c_{1}+3 c_{2}-4 c_{3}+dots dots+ )
( (-1)^{n}(n+1) c_{n}=0 )
(3) ( c_{0}^{2}-c_{1}^{2}+c_{2}^{2}-c_{3}^{2}+dots dots+ )
( (-1)^{n} c_{n}^{2}=0, ) or ( (-1)^{frac{n}{2}} c_{frac{n}{2}} )
according as ( n ) is odd or even.
11
326 The ( 3 r d ) term of ( (2+sqrt{3})^{3} ) is
A . 16
B. 17
c. 18
D. 19
11
327 (a)
J
(0)
1
8.
Coefficient of 124 in (1 +12)12 (1+12) (1 + 24) is (20035)
(a) 12Cg +3 (b) 12Cq+1 (c) 12C ‘(d) 12C7+2
(2004)
21.
1
11
328 Expand:
( left(frac{2}{x}-frac{x}{2}right)^{5} )
11
329 ( [mathrm{AS} 1] ) If ( boldsymbol{A}=frac{1}{3} boldsymbol{B} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{B}=frac{1}{2} boldsymbol{C}, ) then ( boldsymbol{A} )
( B: C= )
A .1: 3: 6
B. 2:3:6
( c cdot 3: 2: 6 )
D. 3: 1: 2
11
330 Using binomial theorem evaluate the
following:
( (98)^{5} )
11
331 Find the middle terms in the expansion
of ( left(2 x^{2}-frac{1}{x}right)^{7} )
В. ( -280 x^{5}, 560 x^{2} )
C ( .560 x^{5},-280 x^{2} )
D . ( 280 x^{5},-560 x^{2} )
11
332 prove that ( C_{0}^{2 n} C_{n}-^{2 n-2} C_{n}+ )
( ^{2 n-4} C_{n}=2^{n} )where( C_{r}=^{n} C_{r} )
11
333 The coefficient of ( 1 / x ) in the expansion of
( (1+x)^{n}(1+1 / x)^{n} ) is
A. ( frac{n !}{(n-1) !(n+1) !} )
B. ( frac{2 n !}{(n-1) !(n+1) !} )
c. ( frac{2 n !}{(2 n-1) !(2 n+1) !} )
D. none of these
11
334 The middle term in the expansion of ( left(1-frac{1}{x}right)^{n}(1-x)^{n}, ) is
A ( cdot^{2 n} C_{n} )
в. ( ^{-2 n} C_{n} )
( mathrm{c} .^{-2 n} C_{n-1} )
D. none of these
11
335 Using the formula for squaring a
binomial the value of ( (999)^{2} ) is:
A. 98009
B. 998005
c. 998001
D. 998002
11
336 Number of rational terms in the
expansion of ( (sqrt{mathbf{2}}+sqrt[4]{mathbf{3}})^{100} ) is
A . 25
B . 26
c. 27
D . 28
11
337 If in the expansion of ( (1+x)^{n}, ) the
coefficients of three consecutive terms
are ( 56,70,56, ) then the value of ( n ) and
the position of the terms of these coefficients are given by
A ( cdot n=8, ) the terms are ( 4^{t h}, 5^{t h}, 6^{t h} )
B . ( n=7 ), the terms are ( 3^{r d}, 4^{t h}, 5^{t h} )
C ( cdot n=8 ), the terms are ( 5^{t h}, 6^{t h}, 7^{t h} )
D. ( n=7 ), the terms are ( 4^{t h}, 5^{t h}, 6^{t h} )
11
338 The number of terms that are integers in the binomial expansion of ( (sqrt{7}+ ) ( sqrt[3]{5})^{35} ) is
A . 4
B. 5
( c cdot 6 )
D.
11
339 If sum of the first 3 coefficients is 16 in
the expansion ( left(x+frac{1}{x^{3}}right)^{n}, ) then find ( n )
A . 10
B. 8
( c .5 )
D.
11
340 The coefficient of ( x^{5} ) in the expansion of
( (x+3)^{8} ) is
A . 1542
в. 1512
( c .2512 )
D. 12
E . 4
11
341 WIL DULU
– 20.
T
The remainder left out when
out when 82n – (62)2n+1 is divided by 9
is:
[2009]
(2) 2 (6) 7 (c) 8 (d) oh
11
342 The coefficients of XP and x9 in the expansion of (1+x)pta
are
[2002]
(a) equal
equal with opposite signs
reciprocals of each other
(d) none of these
11
343 The term independent of ( x ) in the
binomial expansion of ( left(1-frac{1}{x}+3 x^{5}right)left(2 x^{2}-frac{1}{x}right)^{8} )
( mathbf{A} cdot-496 )
B . -400
c. 496
D. 400
11
344 If the second term in the expansion ( left[a^{frac{1}{13}}+frac{a}{sqrt{a^{-1}}}right]^{n} ) is ( 14 a^{5 / 2} ), then the
value of ( frac{n C_{3}}{n C_{2}} ) is
( mathbf{A} cdot mathbf{4} )
B. 3
c. 12
D. 6
11
345 Find the middle term of the expansion of ( left(sqrt{boldsymbol{x}}-frac{mathbf{1}}{boldsymbol{x}}right)^{mathbf{6}} ) 11
346 Find the term independent of ( x ) in the
expansion of
( left(1+x+2 x^{2}right)left(frac{3 x^{2}}{2}-frac{1}{3 x}right)^{9} )
11
347 In the expansion of ( (1+x)^{n} ) the
coefficients of ( p^{t h} ) and ( (p+1)^{t h} ) terms
are respectively ( p ) and ( q ) then ( p+q= )
( mathbf{A} cdot boldsymbol{n} )
B. ( n+1 )
( c cdot n+2 )
( mathbf{D} cdot n+3 )
11
348 The coefficient of ( x ) in the expansion of
( left(1-x-x^{2}+x^{3}right)^{6} ) is ?
( A cdot 6 )
B. -6
c. -12
D. 12
11
349 The fourth term in the expansion of ( left(p x+frac{1}{x}right)^{n} ) is ( frac{5}{2} . ) Then
This question has multiple correct options
( mathbf{A} cdot n=6 )
B . ( n=7 )
c. ( p=frac{1}{2} )
D. ( p=frac{1}{4} )
11
350 If ( A ) is the coefficient of the middle term
in the expansion of ( (1+x)^{2 n} ) and ( B ) and
( mathrm{C} ) are the coefficients of two middle
terms in the expansion of ( (1+x)^{2 n-1} )
then
A. ( A+B=C )
в. ( B+C=A )
c. ( C+A=B )
D. ( A+B+C=0 )
11
351 The total number of rational terms in
the expansion of ( (mathbf{7 3}+mathbf{1 1 9})^{mathbf{6 5 6 1}} )
A . 73
в. 729
( c .728 )
D. 730
E. 732
11
352 The coeffecients of the middle term in
the binomial expansion in powers of ( x ) of ( (1+alpha x)^{4} ) and ( (1+alpha x)^{6} ) is the same
if ( boldsymbol{alpha} ) equals
A ( cdot-frac{5}{3} )
в. ( frac{10}{3} )
( c cdot frac{3}{10} )
D.
11
353 The coefficient of ( x^{n} ) in the expansion of ( frac{1}{(1-x)(1-2 x)(1-3 x)} ) is
A ( cdot frac{1}{2}left(2^{n+2}-3^{n+3}+1right) )
B. ( frac{1}{2}left(3^{n+2}-2^{n+3}+1right) )
c. ( frac{1}{2}left(2^{n+3}-3^{n+2}+1right) )
D. none of these
11
354 The middle term in the expansion of
( (1+x)^{2 n} ) is
A. ( frac{1.3 .5 ldots(2 n-1) 2^{n}}{n !} )
в. ( frac{1.2 .3 ldots(2 n-1) 2^{n} x^{n}}{n !} )
c. ( frac{1.3 .5 ldots(2 n-1) x^{n}}{n !} )
D. ( frac{1.3 .5 ldots .(2 n-1) 2^{n} x^{n}}{n !} )
11
355 Find the term of the expansion of ( (a+ )
( b)^{50} ) which is the greatest in absolute
value if ( |boldsymbol{a}|=sqrt{mathbf{3}}|boldsymbol{b}| )
11
356 1
2 Murks)
The sum of the rational terms in the expansion of
(2 + 31/5,10 is.
(1997 – 2 Marks)
11
357 Fidn the ( 7^{t h} ) term from the end in the
expansion of ( left(9 x-frac{1}{3 sqrt{x}}right)^{18}, x neq 0 )
11
358 1.
The larger of 9950 + 10050 and 10150 is
(1982-2 Marks)
7
11
359 If ( (mathbf{1}+boldsymbol{x})^{mathbf{1 0}}=boldsymbol{a}_{mathbf{0}}+boldsymbol{a}_{mathbf{1}} boldsymbol{x}+boldsymbol{a}_{mathbf{2}} boldsymbol{x}^{mathbf{2}}+ )
( ldots ldots a_{10} x^{10}, ) then value of
( left(a_{0}-a_{2}+a_{4}-a_{6}+a_{8}-a_{10}right)^{2}+ )
( left(a_{1}-a_{3}+a_{5}-a_{7}+a_{9}right)^{2} ) is
( mathbf{A} cdot 2^{10} )
B . 2
( c cdot 2^{20} )
D. None of these
11
360 Find the coefficient of: ( x ) in the
expansion of ( left(1-3 x+7 x^{2}right)(1-x)^{16} )
Enter 1 if answer is -19 otherwise enter
0
11
361 If the coefficients of ( x^{2} ) and ( x^{3} ) are both
zero, in the expansion of the expression ( left(1+a x+b x^{2}right)(1-3 x)^{15} ) in powers of ( x )
then the ordered pair ( (a, b) ) is equal to:
A. (28,315)
B. (-54,315)
c. (-21,714)
D. (24,861)
11
362 The number of integral terms in the expansion of ( (2 sqrt{5}+sqrt[66]{7})^{642} )
A. 105
B. 107
( c .321 )
D. 108
11
363 If the number of terms in the expansion
( (2 x+y)^{n}-(2 x-y)^{n} ) is ( 8, ) then the
value of ( n text { is } ldots ldots . . . . text { (where } n text { is odd }) )
A . 17
B. 19
c. 15
D. 13
11
364 ( (sqrt{2}+1)^{6}+(sqrt{2}-1)^{6}= )
A . 99
B. 98
c. 196
D. 198
11
365 Number of terms in the expansion of ( left(x^{1 / 3}+x^{2 / 5}right)^{40} ) with integral power of ( x )
is equal to
11
366 Compute the summation ( sum_{k=0}^{27} kleft(_{k}^{27}right)left(frac{1}{2}right)^{k}left(frac{2}{3}right)^{27-k} ) 11
367 If the coefficient of the middle term in
the expansion of ( (1+x)^{2 n+2} ) is ( p ) and
the coefficients of middle terms in the
expansion of ( (1+x)^{2 n+1} ) are ( q ) and ( r )
then
A ( . p+q=r )
В. ( p+r=q )
( mathbf{c} cdot p=q+r )
D. ( p+q+r=0 )
11
368 Expand the following binomial ( left(1+frac{x}{2}right)^{7} ) 11
369 Find the cube of the following binomial expressions:
( 4-frac{1}{3 x} )
11
370 If the middle term of ( (1+x)^{2 n} ) is the
greatest term then ( x ) lies between
A. ( n-1<x<n )
в. ( frac{n}{n+1}<x<frac{n+1}{n} )
c. ( n<x<n+1 )
D. ( frac{n+1}{n}<x<frac{n}{n+1} )
11

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